The transfer of binary data by carrier phase shifts, commonly known as binary phase shift keying (BPSK), is a well-known technique and an alternative to amplitude modulation or frequency modulation data transfer techniques. It is however common for these various techniques to be combined, for example, a sub-carrier modulated by phase-shift keying used as a signal for the amplitude modulation or the frequency modulation of a main carrier.
Phase shift keying (“PSK”) modulation is frequently used to transmit digital data. PSK involves shifting the phase of the carrier according to the value of the digital data. For example, in binary PSK (“BPSK”) the “zeros” in the digital data may be represented by a 180 degree shift in the phase of the carrier, while the “ones” in the digital data may be represented by no phase shift. Other degrees of phase shifting may be used. Quadrature PSK (“QPSK”) involves phase shifts of 0, 90, 180 and 270 degrees. PSK typically is referred to as “MPSK” where the “M” represents the number of phases. (The term “M-ary” is also commonly used.)
After a transmitter sends an MPSK signal over the selected transmission medium (e.g., telephone lines or radio frequency waves), a receiver detects the phase changes in the signal. In order to do this accurately, the receiver must extract the unmodulated frequency and phase (commonly referred to as the reference frequency and phase) of the carrier from the received signal. Traditionally, phase-locked loop (“PLL”) circuits have been used to acquire carrier phase. PLLs are relatively easy to implement with either analog or digital technology and, in general, are considered to have good “steady state” performance. Receivers incorporating digital signal processors (DSPs) are well known.
The assessment of the propagation conditions that prevail on a communications link, whether it is a point-to-point terrestrial link or an earth-space link, is of vital importance for the optimal operation of the communications link. The significant indicator of the communications quality of the link is the signal-to-noise ratio (SNR) γ. If the value of this quantity goes below a certain given threshold due, for example, to atmospheric conditions such as rain, the bit error rate on the link becomes unacceptable for reliable communications integrity. Instantaneous (or real-time) knowledge of the dynamic behavior of the SNR γ is thus essential for the optimal implementation of procedures to mitigate the further degradation of γ. In many instances, a separate propagation receiver (or pilot signal) is employed to acquire an associated signal transmitted by a beacon on a communications satellite. Measurement of the fading conditions of this signal level is then extrapolated to that of the operational communications link.
Many methods have been advanced to estimate the SNR using an active modulated communications channels. For example, the output of the receiver matched filter can be sampled, i.e., the voltage level VS for an output symbol, and the value is compared to a pair of a priori determined voltage levels ±α,α>0α<√{square root over (VS)}. The statistical frequency of occurrence nF of values which fail to fall within the interval [−α,α] is calculated as well as the total number nT of samples considered. The ratio nF/nT is related to the symbol error probability by
                    n        F                    n        T              =          2      ⁢              Q        ⁡                  (                                    k              ⁢                                                2                  ⁢                                      E                    S                                                                    N                  0                                                              )                      ,      k    ≡          ⌊              1        -                  α                                    V              S                                          ⌋      where ES is the symbol energy (i.e., energy per symbol), N0 twice the noise power at the output of the matched filter and Q( . . . ) is the well-known Q-function. This relationship is then solved for the SNR γ=ES/N0 by using the inverse function Q−1( . . . ). Although this method has several shortcomings, e.g., limited dynamic range and sensitivity to automatic gain control variations and inter-symbol interference, it suffers from an irreconcilable defect; the inversion of the function Q( . . . ), necessary to obtain the estimate {circumflex over (γ)} of γ, is mathematically correct only if dQ/dγ≠0. In the event that dQ/dγ→0, the problem of determining {circumflex over (γ)} becomes ill-posed. That is, a small error in the estimate of nF/nT leads to a large error in the estimate {circumflex over (γ)}. In fact, the ill-posedness of this problem is the major source of the lack of dynamic range.
As is well-known, for a problem to be well-posed, in the sense of Hadamard, it must meet the following criteria: (1) for each set of data, there exists a solution; (2) the solution is unique; and (3) the solution is stable, i.e., depends continuously on the problem data. If a problem does not meet one or more of these criteria, the problem is considered to be ill-posed. (See, e.g., A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (John Wiley & Sons, New York, 1977).)
An entirely different approach can be imagined which avoids the ill-posedness of the technique described above, and thus tends to be more robust in the presence of measurement errors inherent in the sampling of the matched filter output. In the bi- (or binary) phase-shift keying (BPSK) case, the sampled voltage VS is related to the bi-phase signal amplitude ±A and the corresponding in-phase noise component Nc byVS=K(±A+Nc)where K is a constant proportionality coefficient incorporating receiver gain factors, etc. Remembering that the goal here is to obtain an expression for the SNR γ=ES/N0=A2/2σN2 where σN2=Nc2 for zero mean, white Gaussian noise, the method endeavors to obtain this ratio solely from the measured values of VS. Thus, to separate the noise term, one would try to form the average VS using the fact that Nc=0. However, the random bipolar nature of the signal amplitude A also yields a zero average, giving VS=0. The technique that is then adopted in this approach is to form the absolute value |VS| of each sample and then forming the ensemble average giving
      〈                        V        S                    〉    =            〈                                K          ⁡                      (                                          ±                A                            +                              N                c                                      )                                      〉        ≈    KA  so long as the condition A>>σN prevails. Additionally, the sampled values VS are used to compute the varianceVS2=K2(A2+σN2)
Hence, using the former expression to rid to the A2 term to giveσN2=VS2−−|VS|2 therefore allowing one to write
  γ  =                    A        2                    2        ⁢                                  ⁢                  σ          N          2                      =                            〈                                                V              s                                            〉                2                    2        ⁢                  (                                    〈                              V                s                2                            〉                        -                                          〈                                                                        V                    s                                                                    〉                            2                                )                    solely in terms of the voltage samples VS.
A major drawback of the technique described above is the formation and use of the absolute value of the random quantity VS; such an operation can drastically change the statistical characteristics of the random variable.
Even though using the absolute value of the random quantity VS was done to get rid of the bipolar nature of the communications signal amplitude, as will become evident from an understanding of the techniques of the present invention, as presented hereinbelow, using the absolute value of the random quantity VS will be shown to be mathematically faulty and, moreover, needless. As taught by the present invention (described hereinbelow), one can, and in fact should, incorporate the bipolar nature of the signal amplitude into a rigorous statistical analysis; this characteristic of the signal being just as important as its other aspects. In addition, the aforementioned absolute value approach requires the use of a predetermined bit-stream format within the communications data composed of a series of 1's, thus necessitating a synchronization with, for example, a preamble within the modulation format. This results in a further complication of its implementation and will be shown to be unnecessary.
It is an underlying purpose of the present invention to formulate the rigorous statistical basis for the correct estimation of BPSK signal SNR from what is known about its behavior at the input and output of the receiver demodulator. Instead of employing tacit and unwarranted assumptions concerning the nature of the communication signal for analytical simplification, a complete consistent statistical description of a BPSK signal will be provided to which the well-known techniques of maximum likelihood estimation theory can be applied. By employing, rather than neglecting, all the subtitles of the statistics describing the BPSK signal, an unbiased estimation procedure will be derived that makes simple use of its inherent phase characteristics at the demodulator. In the following description, a preliminary review of BPSK signal representation will be given which will lay the foundation for the statistical connection between Gaussian noise and SNR. Once an appropriate probabilistic description is obtained that establishes a rigorous contact between SNR and the measured phase error of the BPSK signal entering the receiver demodulator, the methods of maximum likelihood estimation theory will be used to obtain analytical expressions for biased and unbiased estimates of SNR from easily measured phase errors. Finally, the straightforward modifications needed at the demodulator to implement the required phase measurements will be given. It should be noted that the resulting SNR estimation technique is also applicable for a quarterary phase-shift keying (QPSK) demodulator simply by applying it to one of the BPSK arms with appropriate modifications for the SNR expression.